Very few people ever object to take an oath or an affirmation – but those that do always do on religious grounds. However, most of the objections are NOT because of a lack of a belief in a god – atheism – but actually because of their religious belief in God!
Those who are atheists typical will take an “affirmation” – and leave out the “so help me God” at the end of the typical Oath. Above and beyond that – the wording is the same.
HOWEVER, people of deep faith like the Quakers and even Hasidic Jews refuse to take an Oath or Affirmation based on the bible…
An objection of someone of the Christian faith might be:
“I say to you: ‘Swear not at all’“…”Above all, my brothers, do not swear—not by heaven or by earth or by anything else. Let your “Yes” be yes, and your “No,” no, or you will be condemned.” The Apostle James: James 5:12
A Christian or Jewish example might come from concept that making a false oath, even unintentionally, would violate a Biblical commandment in Leviticus 19:12
…And ye shall not swear by My name falsely; neither shalt thou profane the name of thy God: I am the LORD